Infinite-dimensional Setting Research Articles

Orthogonality and norm attainment of operators in semi-Hilbertian spaces

We study the semi-Hilbertian structure induced by a positive operator A on a Hilbert space \({\mathbb {H}}.\) Restricting our attention to \(A-\)bounded positive operators, we characterize the norm attainment set and also investigate the corresponding compactness property. We obtain a complete characterization of the \(A-\)Birkhoff–James orthogonality of \(A-\)bounded operators under an additional boundedness condition. This extends the finite-dimensional Bhatia-\( \breve{S} \)emrl Theorem verbatim to the infinite-dimensional setting.

Discrete-time model-based output regulation of fluid flow systems

Model-based discrete-time output regulator design is proposed for fluid flow systems using a geometric approach. More specifically, a class of vortex shedding and falling thin film phenomena modelled by complex Ginzburg–Landau equation (CGLE) and Kuramoto–Sivashinsky equation (KSE) are considered. Differently from a traditional continuous-time controller design, a novel discrete-time modelling technique is proposed in a general infinite-dimensional state-space setting, which does not pertain any spatial approximation or model reduction, and preserves model intrinsic properties (such as stability, controllability and observability). Based on the time discretized plant model (CGLE and KSE systems) by the Cayley–Tustin method, discrete regulator regulation equations are established and facilitated for an output regulator design to achieve fluid flow control and manipulation. To address model instability, a spectrum analysis is utilized in stabilizing continuous-time CGLE and KSE systems, and a link between discrete- and continuous-time closed-loop system stabilizing gains is established. Finally, the proposed methodology is demonstrated through a set of simulation cases, by which the output tracking, disturbance rejection, and model stabilization are achieved for the considered CGLE and KSE systems.

Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then (M, g) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kahler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature.

Finitely additive exchange economies with properness allocation

This is a short and compact survey of finitely additive models in Equilibrium theory; starting from the classical finite-dimensional countably additive model as investigated by Aumann we present the further developments and the core-walras eqivalences that have been proven in finitely additive coalitional finitely dimensional models as well as those in infinite dimensional settings, both in the countably additive case as well as in the finitely additive one.

The infinitesimal generator of the stochastic Burgers equation

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially L^2-ergodic, and that the stochastic Burgers equation on the real line is ergodic.

A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs

We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a semismooth Newton method. This reduces the computational costs in comparison to semismooth Newton methods while maintaining local superlinear convergence. The method applies to Hilbert space problems whose objective is the sum of a smooth function, a regularization term, and a nonsmooth convex function. In the theoretical part of this work we establish the local superlinear convergence of the method in an infinite-dimensional setting and discuss its application to sparse optimal control of the heat equation subject to box constraints. We verify that the assumptions for local superlinear convergence are satisfied in this application and we prove that convergence can take place in stronger norms than that of the Hilbert space if initial error and problem data permit. In the numerical part we provide a thorough study of the hybrid approach on two optimal control problems, including an engineering problem from magnetic resonance imaging that involves bilinear control of the Bloch equations. We use this problem to demonstrate that the new method is capable of solving nonconvex, nonsmooth large-scale real-world problems. Among others, the study addresses mesh independence, globalization techniques, and limited-memory methods. We observe throughout that algorithms based on the hybrid methodology are several times faster in runtime than their semismooth Newton counterparts.

State Convertibility in the von Neumann Algebra Framework

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen's theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of $II_1$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general $II_1$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR.

On the Derivation of Quasi-Newton Formulas for Optimization in Function Spaces

Newton’s method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton’s method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. It is known that quasi-Newton update formulas are solutions to certain variational problems over the space of symmetric matrices. In this paper, we formulate similar variational problems over the space of bounded symmetric operators in Hilbert spaces. By changing the constraints of the variational problem we obtain updates (for the Hessian and Hessian inverse) not only for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method but also for Davidon–Fletcher–Powell (DFP), Symmetric Rank One (SR1), and Powell-Symmetric-Broyden (PSB). In addition, for an inverse problem governed by a partial differential equation (PDE), we derive DFP and BFGS “structured” secant formulas that explicitly use the derivative of the regularization and only approximates the second derivative of the misfit term. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods.

Dichotomy of Linear Partial Differential Equations of Neutral Type in Banach Spaces

This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback theory, we introduce a new transformation of neutral-type equations which plays a key role in our investigation.

Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

We consider a class of linear matrix equations involving semi-infinite matrices which have a quasi-Toeplitz structure. These equations arise in different settings, mostly connected with PDEs or the study of Markov chains such as random walks on bidimensional lattices.We present the theory justifying the existence of the solution in an appropriate Banach algebra which is computationally treatable, and we propose several methods for computing them. We show how to adapt the ADI iteration to this particular infinite dimensional setting, and how to construct rational Krylov methods. Convergence theory is discussed, and numerical experiments validate the proposed approaches.

Connectedness of attractors of a certain family of IFSs

Let X be a Banach space and f,g\colon X\rightarrow X be contractions. We investigate the set C_{f,g}:=\{w\in X\colon\mathrm {the\: attractor\: of\: IFS} \:\mathcal F_w=\{f,g+w\}\: \mathrm {is\: connected}. The motivation for our research comes from papers of Miculescu and Mihail, where it was shown that C_{f,g} is a countable union of compact sets, provided f,g are linear bounded operators with \parallel f\parallel,\parallel g\parallel < 1 and such that f is compact. Moreover, in the case when X is finitely dimensional, such sets have been intensively investigated in the last years, especially when f and g are affine maps. As we will be mostly interested in infinite dimensional spaces, our results can be also viewed as a next step into extending of such studies into infinite dimensional setting. In particular, unlike in the finitely dimensional case, if X has infinite dimension then C_{f,g} is very small set (at least nowhere dense) provided f,g satisfy some natural conditions.

Existence of solutions for a class of multivalued functional integral equations of Volterra type via the measure of nonequicontinuity on the Fréchet space [formula omitted

The existence of continuous not necessarily bounded solutions of nonlinear functional Volterra integral inclusions in infinite dimensional setting is shown with the aid of the measure of nonequicontinuity. New abstract topological fixed point results for admissible condensing operators are introduced. Weak compactness criterion in the space of locally integrable functions in the sense of Bochner is set forth. Some examples illustrating the usefulness of the presented approach are also included.

Padé-type Approximations to the Resolvent of Fractional Powers of Operators

We study a reliable pole selection for the rational approximation of the resolvent of fractional powers of operators in both the finite and infinite dimensional setting. The analysis exploits the representation in terms of hypergeometric functions of the error of the Pade approximation of the fractional power. We provide quantitatively accurate error estimates that can be used fruitfully for practical computations. We present some numerical examples to corroborate the theoretical results. The behavior of rational Krylov methods based on this theory is also presented.

Bayesian inversion for electrical-impedance tomography in medical imaging using the nonlinear Poisson–Boltzmann equation

We develop an electrical-impedance tomography (EIT) inverse model problem in an infinite-dimensional setting by introducing a nonlinear elliptic PDE as a new EIT forward model. The new model completes the standard linear model by taking the transport of ionic charge into account, which was ignored in the standard equation. We propose Bayesian inversion methods to extract electrical properties of inhomogeneities in the main body, which is essential in medicine to screen the interior body and detect tumors or determine body composition. We also prove well-definedness of the posterior measure and well-posedness of the Bayesian inversion for the presented nonlinear model. The new model is able to distinguish between liquid and tissues and the state-of-the-art delayed-rejection adaptive-Metropolis (DRAM) algorithm is capable of analyzing the statistical variability in the measured data in various EIT experimental designs. This leads to design a reliable device with higher resolution images which is crucial in medicine for diagnostic purposes. We first test the validation of the presented nonlinear model and the proposed inverse method using synthetic data on a simple square computational domain with an inclusion. Then we establish the new model and robustness of the proposed inversion method in solving the ill-posed and nonlinear EIT inverse problem by presenting numerical results of the corresponding forward and inverse problems on a real-world application in medicine and healthcare. The results include the extraction of electrical properties of human leg tissues using measurement data.

Operators polynomially isometric to a normal operator

Let H \mathcal {H} be a complex, separable Hilbert space and let B ( H ) \mathcal {B}(\mathcal {H}) denote the algebra of all bounded linear operators acting on H \mathcal {H} . Given a unitarily-invariant norm ‖ ⋅ ‖ u \| \cdot \|_u on B ( H ) \mathcal {B}(\mathcal {H}) and two linear operators A A and B B in B ( H ) \mathcal {B}(\mathcal {H}) , we shall say that A A and B B are polynomially isometric relative to ‖ ⋅ ‖ u \| \cdot \|_u if ‖ p ( A ) ‖ u = ‖ p ( B ) ‖ u \| p(A) \|_u = \| p(B) \|_u for all polynomials p p . In this paper, we examine to what extent an operator A A being polynomially isometric to a normal operator N N implies that A A is itself normal. More explicitly, we first show that if ‖ ⋅ ‖ u \| \cdot \|_u is any unitarily-invariant norm on M n ( C ) \mathbb {M}_n(\mathbb {C}) , if A , N ∈ M n ( C ) A, N \in \mathbb {M}_n(\mathbb {C}) are polynomially isometric and N N is normal, then A A is normal. We then extend this result to the infinite-dimensional setting by showing that if A , N ∈ B ( H ) A, N \in \mathcal {B}(\mathcal {H}) are polynomially isometric relative to the operator norm and N N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A A is normal, while if the spectrum of N N is not of this form, then there always exists a nonnormal operator B B such that B B and N N are polynomially isometric. Finally, we show that if A A and N N are compact operators with N N normal, and if A A and N N are polynomially isometric with respect to the ( c , p ) (c,p) -norm studied by Chan, Li, and Tu, then A A is again normal.

Bayesian Inference and Uncertainty Quantification for Medical Image Reconstruction with Poisson Data

We provide a complete framework for performing infinite dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second, we provide a dimension-independent Markov chain Monte Carlo algorithm, based on the preconditioned Crank--Nicolson Langevin method, in which we use a primal-dual scheme to compute the offset direction. Third, we give a method combining the model discrepancy method and maximum likelihood estimation to determine the regularization parameter in the hybrid prior. Finally we propose to use the obtained posterior distribution to detect artifacts in a recovered image. We provide an example to demonstrate the effectiveness of the proposed method.

Error Estimates of Penalty Schemes for Quasi-Variational Inequalities Arising from Impulse Control Problems

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton--Jacobi--Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the scheme is half-order accurate for HJBQVIs with Lipschitz coefficients and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite-dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.

Deep Neural Networks Motivated by Partial Differential Equations

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of multivariate functions and the output of image processing algorithms as solutions to certain PDEs. Posing image processing problems in the infinite-dimensional setting provides powerful tools for their analysis and solution. For the last few decades, the reinterpretation of classical image processing problems through the PDE lens has been creating multiple celebrated approaches that benefit a vast area of tasks including image segmentation, denoising, registration, and reconstruction. In this paper, we establish a new PDE interpretation of a class of deep convolutional neural networks (CNN) that are commonly used to learn from speech, image, and video data. Our interpretation includes convolution residual neural networks (ResNet), which are among the most promising approaches for tasks such as image classification having improved the state-of-the-art performance in prestigious benchmark challenges. Despite their recent successes, deep ResNets still face some critical challenges associated with their design, immense computational costs and memory requirements, and lack of understanding of their reasoning. Guided by well-established PDE theory, we derive three new ResNet architectures that fall into two new classes: parabolic and hyperbolic CNNs. We demonstrate how PDE theory can provide new insights and algorithms for deep learning and demonstrate the competitiveness of three new CNN architectures using numerical experiments.

A continuous selection for optimal portfolios under convex risk measures does not always exist

Risk control is one of the crucial problems in finance. One of the most common ways to mitigate risk of an investor’s financial position is to set up a portfolio of hedging securities whose aim is to absorb unexpected losses and thus provide the investor with an acceptable level of security. In this respect, it is clear that investors will try to reach acceptability at the lowest possible cost. Mathematically, this problem leads naturally to considering set-valued maps that associate to each financial position the corresponding set of optimal hedging portfolios, i.e., of portfolios that ensure acceptability at the cheapest cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note, we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.

Alpha-Beta Log-Determinant Divergences Between Positive Definite Trace Class Operators

This work presents a parametrized family of divergences, namely Alpha-Beta Log-Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highly general and contains many divergences as special cases, including the recently formulated infinite-dimensional affine-invariant Riemannian distance and the infinite-dimensional Alpha Log-Det divergences between positive definite unitized trace class operators. In particular, it includes a parametrized family of metrics between positive definite trace class operators, with the affine-invariant Riemannian distance and the square root of the symmetric Stein divergence being special cases. For the Alpha-Beta Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices.